In this chapter, the effect of an external static electric field is described within the BenDa niel-Duke model for the conduction band, introduced in the previous chapter. In Section 4

4.1. Introduction
Application of an electric field introduces a difference in chemical potential between the contact regions of the DBRT structure, that serve as reservoirs, and hence causes an electric current to flow through the device. Charge displacement induces an additional field, which poses the problem of selfconsistency. The selfconsistent field was shown to be responsible for a bistable or tristable current. Experimental evidence for this predicted phenomenon was not considered convincing, until a simultaneous monitoring of the charge build-up in the DBRT structure was performed.

In this chapter, the effect of an external static electric field is described within the BenDa.niel-Duke model for the conduction band, introduced in the previous chapter. In Section 4.2 the effect of charge build-up in the doped contact layers is studied. This space charge is of importance mainly to the voltage axis of the I-V characteristic, although the current scale is affected also. The charge build-up in the well is the topic of Section 4.3, in which its relation to the current bistability is clarified.

4.2. The accumulation and depletion layer

The electric field in the DBRT structure due to the applied bias voltage is accompanied by space charge in the doped contact regions to the left and the right of the central intrinsic layers. Generally, a charge sheet of zero width is assumed, yielding a constant field inside and no field outside the central structure. Such a potential profile is reasonable for structures having heavily doped electrodes that extend up to the barriers. In the case of moderately doped electrodes or undoped spacer layers, it does not apply. Screening lengths on either side of the central layers are then to be introduced, A more realistic charge distribution would extend into the doped layers, causing substantial band bending according to Poisson's equation:

(4.1) where is the conduction band minimum, and is the static dielectric constant that may be different in layers of different material ( for material A, B). is the charge density that can be written as:
(4.2)
where is the doping profile, and the density of conduction band electrons. The boundary conditions for (4.1) are . Since the total structure must remain charge neutral, an additional restriction is obtained from:
(4.3)
which reads in terms of the electric displacement F(z) the electric field:
.
To solve (4.1) we need an expression for the electron density . An approximate expression for can be borrowed from the well-known ThomasFermi screening theory, generalizing the equilibrium expression:
(4.4)

valid for constant to cases where varies slowly with position. Let us denote the well and barrier widths by and , and choose to be the middle of the well. Writing:
for or (4.5a)
and zero otherwise, and:
(4.5b)

where the chemical potentials are given by:

we can solve the system (4.1-2) numerically. There are, however, many reasons (to be mentioned later) to not take this solution very seriously, especially for large bias . Here, we will present only a drastic simplification.
In this simplification, we do not link the functions n(z) and to each other, but their average values and in the accumulation layer, to the left of the first reservoir, and and in the depletion layer to the right of the second reservoir. Now (4.5b) becomes:

(4.6)

and zero otherwise. In (4.6), and are the widths of the accumulation and depletion .layer, to be determined by a selfconsistent solution of the system. Subsituting (4.6) in Poisson's equation (4.1), we obtain the potential drops in both charge layers:

(4.7)

Charge neutrality (4.3) yields:
(4.8)
The electric displacement D in the central layers inside the "capa.citor11 is: , so that the voltage drop across the well and barriers is found to be:

The total voltage drop across the total structure including the electrodes must equal the applied bias :voltage, which boundary condition reads:
(4.9)
Eqs.(4.6-9) constitute a system of six equations with six unknown quantities: , , and , , , that can hence be solved. The lengths and are of the order of the Debye or Thomas-Fermi screening length, as can be seen as follows. For the accumulation layer, Eqs.(4.1-4) can be combined into a single second-order, non-linear differential equation:


where we have made use of the property of the Fermi-Dirac integrals, that . We can formally solve this equation by substituting:

which automatically satisfies one boundary condition: . This leaves one parameter, , to be determined by the other boundary condition, while all other coefficients are recursi:vely related to
:
The inverse screening length q0 is found to equal:
(4.10)
which for high temeratures approximates the inverse Debye length:

and for low temperatures reduces to the well-known Thomas-Fermi expression:

where is the Fermi wave number , and is the effective Bohr radius. Hence for la.rge negative z the band minimum decays exponentially over a length For small , we find for the system (4.6-9) that:

Thus is a decreasing, an increasing function of .

In the same way, the average potential in the accumulation layer is found to be:
(4.11) We now introduce an effective Fermi energy by writing (4.6) for the accumulation layer as . Hence . Through the last term, the effective Fermi energy will depend on the applied bias. For small , we have, using (4.11):